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arXiv:quant-ph/0509188 · Physical Review A

Efficient implementation of controlled rotations by using entanglement

Ming-Yong Ye, Yong-Sheng Zhang, Guang‐Can Guo

Published Mar 27, 2006• 19 citations
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Abstract

Implementation of controlled rotations using entanglement is considered. We use a general two-qubit pure state to implement controlled rotations on two separate qubits. The probability of the success we achieve is higher than the known results in some parameter regions. Qualitatively the probability of the success we achieve is an increasing function of the entanglement and a decreasing function of the controlled-rotation angle, and the probability of success will approach a unit when the entangled state approaches a Bell state or the controlled-rotation angle approaches zero.

DOI
10.1103/physreva.73.032337

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Implementation of controlled rotations using entanglement is considered. We use a general two-qubit pure state to implement controlled rotations on two separate qubits. The probability of the success we achieve is higher than the known results in some parameter regions. Qualitatively the probability of the success we achieve is an increasing function of the entanglement and a decreasing function of the controlled-rotation angle, and the probability of success will approach a unit when the entangled state approaches a Bell state or the controlled-rotation angle approaches zero.
The nonlocal correlations between quantum states in an entangled system are essential to many quantum communications applications. A basic quantum operation, which permits the distribution of entanglement between two initially uncorrelated systems, is entanglement swapping. Here we present a rigorous formulation of entanglement swapping of any two partially mixed two-qubit states without limiting ourselves to any specific type of state or noise. Further, for two important classes of the input states, Bell diagonal and pure states, we describe how the concurrence of the final state is related to the concurrence of the initial states. First, we consider Bell diagonal states, and find bounds on the concurrence of the final state in terms of the concurrences of the initial states. These bounds are important for communications applications because polarization mode dispersion in fibers produces Bell diagonal states up to a local unitary rotation. Second, we show that swapping pure states occasionally results in a state of higher concurrence than either of the initial states. In addition, we find that two pure states are most likely to be capable of swapping to a state of increased concurrence when the two initial states have similar concurrences. Our analysis offers a completely general framework for investigating the behavior of any pair of two-qubit states when used for entanglement swapping.
Teleportation of quantum gates using partially entangled states is considered. Different from the known probability schemes, we propose and study a method for teleporting a prototypical single-qubit rotation on a remote receiver with unit fidelity and unit probability by using two partially entangled pairs. The method is applicable to any two partially entangled pairs satisfying the condition that their smaller Schmidt coefficients γ and η are (2γ+2η-2γη-1)≥0. In our scheme, the sender's local generalized measurement described by a positive operator-valued measurement (POVM) lies at the heart. We construct the required POVM. The fact that the controlled teleportation of single-qubit rotation could be realized exactly using two partially entangled pairs is also notable. A sender could teleport a rotation on a remote receiver, an arbitrary one of the two receivers, via the control of the other in a network.
We show how a many-to-one and a one-to-many controlled remote quantum rotations (CRQRs) can be implemented deterministically and exactly by using partially entangled quantum channels. Firstly, we present a theoretical scheme for a N -to-1 CRQR, in which the quantum rotations initially distributed in N spatially separated qubits can be exactly and deterministically performed onto a remote single qubit via two partially entangled ( N + M +1)-qubit Greenberger-Horne-Zeilinger (GHZ) states without performing any global operations. The feature of this scheme is that, apart from N senders and a receiver, M agents are included in the process as controllers. Should any one of the M agents not cooperate, the receiver could not gain the original rotations. Then, we design another scheme for implementing a 1-to- N CRIC with unit fidelity and unit probability by employing a partially entangled (2 M + N +1)-qubit Einstein-Podolsky-Rosen (EPR)-GHZ state or K partially entangled ( M +2)-qubit GHZ states, in which a quantum rotation can be divided into N pieces ( N K ) and performed from a sender onto N distant receivers via the control of M agents in a quantum network. In these schemes, the senders (or the receivers, or the controllers) local positive operator valued measurement (POVM) lies at the heart. We construct the required POVMs. The fact that deterministic and exact implementation of a many-to-one or a one-to-many CRQR could be realized using partially entangled quantum channel is notable. These schemes can be used to quantum secret sharing, quantum voting, and so on. They definitely have the strong security.
Abstract Quantum entanglement is essential to the development of quantum computation, communications, and technology. The controlled SWAP test, widely used for state comparison, can be adapted to an efficient and useful test for entanglement of a pure state. Here we show that the test can evidence the presence of entanglement (and further, genuine n-qubit entanglement), can distinguish entanglement classes, and that the concurrence of a two-qubit state is related to the test’s output probabilities. We also propose a multipartite measure of entanglement that acts similarly for n-qubit states. The average number of copies of the test state required to detect entanglement decreases for larger systems, to four on average for many (n ≳ 8) qubits for maximally entangled states. For non-maximally entangled states, the number of copies required to detect entanglement increases with decreasing entanglement. Furthermore, the results are robust to second order when typical small errors are introduced to the state under investigation.
The quantum Fisher information of a quantum state with respect to a certain parameter quantifies the sensitivity of the quantum state to changes in that parameter. Maximizing the quantum Fisher information is essential for achieving the optimal estimation precision of quantum sensors. A typical quantum sensor involves a qubit(e.g. a spin-1/2) probe undergoing an unknown rotation, here the unknown rotation angle is the parameter to be estimated. A well known limitation is that if the rotation axis is unknown, the maximal quantum Fisher information is impossible to attain. This limitation has been lifted recently by leveraging entanglement between the probe qubit and an ancilla qubit. Namely, through measurement of the ancilla after the axis is revealed, one can prepare the probe that is optimal for any unknown rotation axis. This proposal, however, works only for a spin-1/2. Considering large spin probes can achieve a larger quantum Fisher information, offering enhanced metrological advantage, we here utilize the entanglement between a large spin probe and an ancilla to achieve optimal quantum Fisher information for estimating the rotation angle, without prior knowledge of the rotation axis. Different from the previous spin-1/2 case, achieving the optimal precision with large spins generally requires post-selection, resulting in a success probability dependent on the dimension of the Hilbert space. Furthermore, we extend the encoding state from the maximally entangled case to general entangled states, showing that optimal metrology can still be achieved with a certain success probability.